Annals of Global Analysis and Geometry

, Volume 46, Issue 3, pp 241–257 | Cite as

Classification of Möbius homogenous surfaces in \({\mathbb {S}}^4\)

  • Changping Wang
  • Zhenxiao XieEmail author


A surface \(M^2\) in \(\mathbb {S}^4\) is called Möbius homogeneous if for any two points \(p,q \in M^2\) there exists a Moebius transformation which takes \(p\) to \(q\) and keeps \(M^2\) invariant. In this paper we give a complete classification of the Möbius homogenous surfaces in \(\mathbb {S}^4\).


Moduli space Möbius invariant Conformal Gauss Map  Möbius homogenous surface 

Mathematics Subject Classification (2000)

53A30 53A55 53C42 



This work is funded by the Project 11171004 and 11331002 of National Natural Science Foundation of China. We thank Professor Haizhong Li and Professor Faen Wu for the helpful discussions and valuable advices.


  1. 1.
    Burstall, F., Ferus, D., Leschke, K., Pedit, F., Pinkall, U.: Conformal geometry of surfaces in the 4-sphere and Quaternions. In: Lect. Notes Math., vol. 1772. Springer, Berlin (2002)Google Scholar
  2. 2.
    Cheng, Q., Shu, S.: A möbius characterization of submanifolds. J. Math. Soc. Jpn. 58, 904–925 (2006)MathSciNetGoogle Scholar
  3. 3.
    Fan, L., Lü, Y., Wang, C., Zhong, J.: Geodesics on the moduli space of oriented circles in \(\mathbb{s}^{3}\). Results Math. 59(3–4), 471–484 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Guo, Z., Li, H., Wang, C.: The moebius characterizations of willmore tori and veronese submanifolds in the unit sphere. Pac. J. Math. 241(2), 227–242 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Guo, Z., Li, H., Wang, C.: The second variational formula for willmore submanifolds. Results Math. 40, 205–225 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hu, Z., Li, H.: Submanifolds with constant Moebius scalar curvaturein \(S^n\). Manuscr. Math. 111, 287–302 (2003)Google Scholar
  7. 7.
    Li, H., Wang, C., Wu, F.: Surfaces with vanishing moebius form in \(s^n\). Acta Math. Sinica. 19, 671–678 (2003)Google Scholar
  8. 8.
    Li, T., Ma, X., Wang, C.: Möbius homogenous hypersurfaces with two distinct principle curvature in \(\mathbb{S}^{n+1}\). Arkiv for Matematik (2013, to appear). doi: 10.1007/s11512-011-0161-5
  9. 9.
    Liu, H., Wang, C., Zhao, G.: Möbius isotropic submanifolds in \(\mathbb{ s}^n\). Tohoku Math. J. 53, 553–569 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ma, X., Wang, C.: Willmore surfaces of constant möbius curvature. Ann. Global Anal. Geom. 32(3), 297–310 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Sulanke, R.: Möbius geometry V: homogeneous surfaces in the Möbius space \(\mathbb{S}^3\). In: Topics in Differential Geometry (Debrecen), vol. 2, pp. 1141C1154 (1984). Colloq. Math. Soc. János Bolyai, vol. 46. North-Holland, Amsterdam (1988)Google Scholar
  12. 12.
    Wang, C.: Möbius geometry of submanifolds in \(\mathbb{s}^n\). Manuscr. Math. 96, 517–534 (1998)CrossRefzbMATHGoogle Scholar
  13. 13.
    Wang, C.: Möbius geometry for hypersurfaces in \(\mathbb{r}^4\). Nagoya Math. J. 139, 1–20 (1995)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.School of Mathematics and Computer ScienceFujian Normal UniversityFuzhouPeople’s Republic of China
  2. 2.School of Mathematical SciencesPeking UniversityBeijingPeople’s Republic of China

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